In this presentation I will review several types of eigenvalues that have been discussed in inverse electromagnetic scattering theory.  The most fundamental, transmission eigenvalues, were first discussed for the Helmholtz equation in the mid 1980s.  They resisted mathematical analysis until 1990, and have since been linked to non-scattering wave numbers and target signatures for inverse problems.  In principle these eigenvalues can be detected from multi-frequency, multistatic,far field scattering.
To avoid the need for multi-frequency data, a new class of eigenvalues called modified transmission eigenvalues was suggested, an example being Steklov eigenvalues which I will present.  The talk will end with a discussion of eigenvalues that can be used to study thin scatterers made of meta-materials that can exhibit anisotropic surface impedance.